“I know“ or “I do know”
- I know that.
- I do know that.
Both versions of this sentence are correct. The second is an emphatic version of the sentence. When we want to give a sentence positive emphasis, when we want to emphasise that it's true, we usually stress the auxiliary verb:
- I can speak French.
- I will do my homework.
- I have been to France.
In present simple and past simple sentences in English there is no auxiliary verb:
- I study hard
- She gave the money to him.
- I know.
If we want to have an emphatic version of these sentences, we need to insert the dummy auxiliary DO. This is the same auxiliary we use to make questions and negatives in the present and past simple. Although in writing we cannot hear the stress, we know that the sentence is emphatic because it has an auxiliary verb:
- I do study hard.
- She did give the money to him.
- I do know.
When we say I do know, this will often sound as if we are contrasting the sentence with some negative idea. This idea might be that the listener thinks that you don't know. Or it might be that you don't know some other information, but you do know this information:
- A: You don't know, do you?
- B: I do know!
or
- I don't know exactly when he did it, but I do know that he did it before he left London.
It is not a good idea to use do in sentences unless you do actually want to give the sentence positive emphasis!
As has previously been said, yes, it's correct.
Other languages may not have this. French and German certainly don't. "Je parle Francais" could mean "I speak French", "I can speak French", or even "I am (currently) speaking French". Context matters for that.
But those same languages both have words - "si" in French, "doch" in German - which translate into English something like "oh yes I do". So if someone says you can't speak French, your reply would be "si, je parle Francais" to contradict them.
One difference you may find is whether it's "English English". In Britain, you would be more likely to hear word stress used for emphasis - you're less likely to hear "I do speak French". And even more so, you would be very unlikely to hear "I do not speak French" - you would hear "I don't speak French". To Brits, the "do not" phrasing instead of "don't" sounds very forceful, as if you were seriously offended by the idea that you might speak French! :) In the US though, this is a normal phrasing and has no such implications.